Capabilities

NoLimits.jl provides a broad set of modeling, estimation, and diagnostic capabilities for nonlinear mixed-effects analysis. All features listed below are implemented and tested in the current package.

Modeling

• Nonlinear mixed-effects models for longitudinal data, with one or multiple random-effect grouping structures (e.g., subject-level and site-level variability in a single model).
• Fixed-effects-only models for settings where random effects are not needed.
• ODE-based and non-ODE models within the same modeling framework: algebraic structural models and mechanistic ODE systems share the same specification language.
• Multiple outcomes in one model, including mixed outcome types (e.g., continuous and count outcomes jointly).
• Hidden Markov outcome models via DiscreteTimeDiscreteStatesHMM and ContinuousTimeDiscreteStatesHMM, enabling latent-state-dependent observation processes. Dedicated parameter types ProbabilityVector, DiscreteTransitionMatrix, and ContinuousTransitionMatrix provide AD-compatible, automatically constrained representations of initial-state distributions and transition or rate matrices.
• Left-censored and interval-censored observations through censored(...) in the observation model.

Random-Effects Flexibility

• Univariate and multivariate random effects.
• Multiple grouping columns, including row-varying non-primary_id group membership for non-ODE and discrete-time HMM models.
• Random-effect distributions beyond the Gaussian family: heavy-tailed (TDist), skewed (SkewNormal), positive-valued (LogNormal, Gamma), and other distributions from Distributions.jl.
• Flow-based random effects via NormalizingPlanarFlow for highly flexible latent distributions.
• Random-effect distributions parameterized by covariates, neural networks, soft decision trees, or spline functions – enabling covariate-dependent heterogeneity that goes beyond standard variance models.

Machine-Learning Integration

• Neural-network parameter blocks (NNParameters) can be embedded in formulas, ODE dynamics, initial conditions, and random-effect distribution parameterizations. Neural-ODE-style models arise naturally when learned components appear inside @DifferentialEquation.
• Soft decision tree parameter blocks (SoftTreeParameters) provide an alternative differentiable function approximator with the same integration points as neural networks.
• Spline parameter blocks (SplineParameters) for smooth, learnable basis-function expansions.
• All learned components can be used at the population level (fixed effects only) or individualized through full-parameter random effects.

Covariate Handling

• Time-varying covariates that change across observations within an individual.
• Constant covariates that are fixed within a grouping level (e.g., baseline age, treatment arm).
• Dynamic (interpolated) covariates that provide continuous-time access within ODE integration, with support for eight interpolation methods from DataInterpolations.jl.

Estimation Methods

All methods share a unified fit_model interface, allowing direct comparison of estimation approaches on the same model and data.

Uncertainty Quantification

• Wald-based intervals from Hessian or sandwich covariance estimates.
• Profile-likelihood intervals via LikelihoodProfiler.jl.
• Posterior-draw intervals from MCMC chains or VI variational posteriors (direct or refit).
• A unified compute_uq interface across all backends.

Diagnostics and Visualization

• Fitted-trajectory plots overlaid on observed data.
• Visual predictive checks (VPCs).
• Residual diagnostics: QQ plots, PIT histograms, autocorrelation plots.
• Random-effects diagnostics: marginal density plots, pairwise scatter, standardized EBE distributions.
• Observation-distribution plots showing full predictive distributions at selected data points.
• Uncertainty distribution plots for estimated parameters.
• Multistart objective waterfalls and parameter stability summaries.

Composability

A defining feature of NoLimits.jl is that the capabilities above are freely composable. A single model can simultaneously use ODE dynamics, multiple learned function approximators, several random-effect grouping levels with non-Gaussian distributions, covariates at different temporal resolutions, and multiple outcome types. This composability is central to the package design and is exercised throughout the test suite.